Page:The Meaning of Relativity - Albert Einstein (1922).djvu/99

LECTURE IV

THE GENERAL THEORY OF RELATIVITY
(Continued)

We are now in possession of the mathematical apparatus which is necessary to formulate the laws of the general theory of relativity. No attempt will be made in this presentation at systematic completeness, but single results and possibilities will be developed progressively from what is known and from the results obtained. Such a presentation is most suited to the present provisional state of our knowledge.

A material particle upon which no force acts moves, according to the principle of inertia, uniformly in a straight line. In the four-dimensional continuum of the special theory of relativity (with real time co-ordinate) this is a real straight line. The natural, that is, the simplest, generalization of the straight line which is plausible in the system of concepts of Riemann's general theory of invariants is that of the straightest, or geodetic, line. We shall accordingly have to assume, in the sense of the principle of equivalence, that the motion of a material particle, under the action only of inertia and gravitation, is described by the equation,

${\frac {d^{2}x_{\mu }}{ds^{2}}}+\Gamma _{\alpha \beta }^{\mu }{\frac {dx_{\alpha }}{ds}}{\frac {dx_{\beta }}{ds}}=0.$

(90)

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